Use the formula for the sum of a geometric sequence to write the following sum in closed form. 83 +84 +85+. +8k, where k is any integer with k 2 3.
Use the formula for the sum of a geometric sequence to write the following sum in closed form. 83 +84 +85+. +8k, where k is any integer with k 2 3.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Topic: Geometric Sequences**
**Problem:**
Use the formula for the sum of a geometric sequence to write the following sum in closed form.
\[ 8^3 + 8^4 + 8^5 + \ldots + 8^k \]
where \( k \) is any integer with \( k \geq 3 \).
**Explanation:**
The sequence given is a geometric sequence with the first term \( a = 8^3 \) and a common ratio \( r = 8 \). The sum of the first \( n \) terms of a geometric sequence can be expressed as:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Here, we need to find the sum of terms from \( 8^3 \) to \( 8^k \). The number of terms in this sequence is \( (k - 3 + 1) = k - 2 \). Therefore, the sum can be expressed using the formula for a geometric sequence sum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2F4523944b-7c64-44a1-81aa-21dc7fe21aae%2Fanckh9q_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Geometric Sequences**
**Problem:**
Use the formula for the sum of a geometric sequence to write the following sum in closed form.
\[ 8^3 + 8^4 + 8^5 + \ldots + 8^k \]
where \( k \) is any integer with \( k \geq 3 \).
**Explanation:**
The sequence given is a geometric sequence with the first term \( a = 8^3 \) and a common ratio \( r = 8 \). The sum of the first \( n \) terms of a geometric sequence can be expressed as:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Here, we need to find the sum of terms from \( 8^3 \) to \( 8^k \). The number of terms in this sequence is \( (k - 3 + 1) = k - 2 \). Therefore, the sum can be expressed using the formula for a geometric sequence sum.
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